Analysis and Design of Algorithm

B. Tech. (CSE), VII Semester

Roshan Tailor

Amity University Rajasthan

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This Lecture

• Single-source shortest paths in weighted graphs– Shortest-Path Problems– Properties of Shortest Paths, Relaxation– Dijkstra’s Algorithm– Bellman-Ford Algorithm

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Shortest Path

• Generalize distance to weighted setting• Digraph G = (V,E) with weight function W: E R

(assigning real values to edges)• Weight of path p = v1 v2 … vk is

• Shortest path = a path of the minimum weight• Applications

– static/dynamic network routing– robot motion planning– map/route generation in traffic

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( ) ( , )k

i ii

w p w v v

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Shortest-Path Problems

• Shortest-Path problems– Single-source (single-destination). Find a shortest

path from a given source (vertex s) to each of the vertices.

– Single-pair. Given two vertices, find a shortest path between them. Solution to single-source problem solves this problem efficiently, too.

– All-pairs. Find shortest-paths for every pair of vertices. Dynamic programming algorithm.

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Negative Weights and Cycles?

• Negative edges are OK, as long as there are no negative weight cycles (otherwise paths with arbitrary small “lengths” would be possible)

• Shortest-paths can have no cycles (otherwise we could improve them by removing cycles)– Any shortest-path in graph G can be no longer than

n – 1 edges, where n is the number of vertices

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Relaxation

• For each vertex v in the graph, we maintain v.d(), the estimate of the shortest path from s, initialized to at the start

• Relaxing an edge (u,v) means testing whether we can improve the shortest path to v found so far by going through u

u v

vu

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Relax(u,v)

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Relax(u,v)

Relax (u,v,G)if v.d > u.d()+ w(u,v) then

v.d = (u.d()+ w(u,v))

v.setparent = u

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Dijkstra's Algorithm

• Non-negative edge weights• Greedy, similar to Prim's algorithm for MST• Like breadth-first search (if all weights = 1, one

can simply use BFS)• Use Q, a priority queue ADT keyed by v.d()

(BFS used FIFO queue, here we use a PQ, which is re-organized whenever some d decreases)

• Basic idea– maintain a set S of solved vertices– at each step select "closest" vertex u, add it to S, and

relax all edges from u

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Dijkstra’s Pseudo Code

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Dijkstra’s Example

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Dijkstra’s Example (2)

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Dijkstra’s Example (3)

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Dijkstra’s Running Time

• Extract-Min executed |V| time• Decrease-Key executed |E| time• Time = |V| TExtract-Min + |E| TDecrease-Key

• T depends on different Q implementations

Q T(Extract-Min)

T(Decrease-Key) Total

array (V) (1) (V 2)

binary heap (lg V) (lg V) (E lg V)

Fibonacci heap (lg V) (1) (amort.) (V lgV + E)

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Bellman-Ford Algorithm

• Dijkstra’s doesn’t work when there are negative edges:– Intuition – we can not be greedy any more on the

assumption that the lengths of paths will only increase in the future

• Bellman-Ford algorithm detects negative cycles (returns false) or returns the shortest path-tree

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Bellman-Ford Algorithm

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Bellman-Ford Example

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Bellman-Ford Example

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• Bellman-Ford running time:– (|V|-1)|E| + |E| = (VE)

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Bellman-Ford Example

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